Affiliation:
1. Department of Mathematics , US Coast Guard Academy , New London , CT USA 06320
2. Department of Mathematics , Xavier University of Louisiana , New Orleans, LA USA 70125
Abstract
Abstract
We show the existence of unbounded solutions to difference equations of the form
{
x
n
+
1
=
c
′
n
x
n
B
n
y
n
,
y
n
+
1
=
b
n
x
n
+
c
n
y
n
A
n
+
C
n
y
n
f
o
r
n
=
0
,
1
,
…
,
\left\{ {\matrix{{{x_{n + 1}} = {{{{c'}_n}{x_n}} \over {{B_n}{y_n}}},} \hfill \cr {{y_{n + 1}} = {{{b_n}{x_n} + {c_n}{y_n}} \over {{A_n} + {C_n}{y_n}}}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots ,
where
{
c
′
n
}
n
=
0
∞
\left\{ {{{c'}_n}} \right\}_{n = 0}^\infty
,
{
B
′
n
}
n
=
0
∞
\left\{ {{{B'}_n}} \right\}_{n = 0}^\infty
,
{
b
n
}
n
=
0
∞
\left\{ {{b_n}} \right\}_{n = 0}^\infty
,
{
c
n
}
n
=
0
∞
\left\{ {{c_n}} \right\}_{n = 0}^\infty
, and
{
A
n
}
n
=
0
∞
\left\{ {{A_n}} \right\}_{n = 0}^\infty
are all bounded above and below by positive constants, and
{
C
n
}
n
=
0
∞
\left\{ {{C_n}} \right\}_{n = 0}^\infty
is either bounded above and below by positive constants or is identically zero. In the latter case, we give an example which can be reduced to a system of the form
{
x
n
+
1
=
x
n
y
n
,
y
n
+
1
=
x
n
+
γ
n
y
n
f
o
r
n
=
0
,
1
,
…
,
\left\{ {\matrix{ {{x_{n + 1}} = {{{x_n}} \over {{y_n}}},} \hfill \cr {{y_{n + 1}} = {x_n} + {\gamma _n}{y_n}} \hfill \cr } \,\,\,\,\,for} \right.\,\,\,n = 0,1, \ldots ,
where 0 < γ′ < γ
n
< γ < 1 for some constants γ and γ′ for all n. This provides a counterexample to the main result of the 2021 paper by Camouzis and Kotsios.
Subject
Applied Mathematics,Numerical Analysis,Statistics and Probability,Analysis
Reference10 articles.
1. [1] E. Camouzis, M. R. S. Kulenović, G. Ladas, and O. Merino. Rational systems in the plane. J. Difference Equ. Appl., 15(3):303–323, 2009.
2. [2] A. M. Amleh, E. Camouzis, G. Ladas, and M. A. Radin. Patterns of boundedness of a rational system in the plane. J. Difference Equ. Appl., 16(10):1197–1236, 2010.
3. [3] E. Camouzis, E. Drymonis, and G. Ladas. Patterns of boundedness of the rational system
xn+1=α1+β1xnA1+B1xn+C1yn
{x_{n + 1}} = {{{\alpha _1} + {\beta _1}{x_n}} \over {{A_1} + {B_1}{x_n} + {C_1}{y_n}}}
and
yn+1=α2+β2xn+γ2ynA2+B2xn+C2yn
{y_{n + 1}} = {{{\alpha _2} + {\beta _2}{x_n} + {\gamma _2}{y_n}} \over {{A_2} + {B_2}{x_n} + {C_2}{y_n}}}
. Comm. Appl. Nonlinear Anal., 18(1):1–23, 2011.
4. [4] Elias Camouzis. Boundedness of solutions of a rational system. Int. J. Difference Equ., 7(1):1–18, 2012.
5. [5] E. Camouzis and G. Ladas. When does periodicity destroy boundedness in rational equations? J. Difference Equ. Appl., 12(9):961–979, 2006.